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\author{Anton}
\begin{document}
$$ S_{n,m} = \sum_{k=0}^n k^m $$
$$ S_{n+1,m} = \sum_{k=0}^{n+1} k^m = \sum_{k=0}^n (k+1)^m $$
$$ S_{n+1,2} = S_{n,2} + (n+1)^2 = \sum_{k=0}^n (k+1)^2 $$
$$ \sum_{k=0}^n (k+1)^2 = \sum_{k=0}^n k^2 + 2k + 1 $$
$$ S_{n,2} + (n+1)^2 = \sum_{k=0}^n k^2 + 2\sum_{k=0}^nk + \sum_{k=0}^n 1 $$
$$ S_{n,2} + (n+1)^2 = S_{n,2} + 2S_{n,1} + (n + 1) \Rightarrow $$
$$ \boxed{ S_{n,1} = \sum_{k=0}^n k = \frac{n(n + 1)}{2} } $$
$$ S_{n,3} + (n + 1)^3 = \sum_{k=0}^n (k + 1)^3 = \sum_{k=0}^n k^3 + 3k^2 + 3k + 1 $$
$$ S_{n,3} + (n + 1)^3 = S_{n,3} + 3S_{n,2} + 3\frac{n(n + 1)}{2} + (n + 1) \Rightarrow $$
$$ \boxed{ S_{n,2} = \frac{n(n + 1)(n + \frac{1}{2})}{3} = \frac{n(n + 1)(2n + 1)}{6} } $$
$$ S_{n,m} + (n+1)^m = \sum_{k=0}^n (k + 1)^m = \sum_{k=0}^n k^m + C_n^1 k^{m-1} + C_n^2 k^{m-2} + ... + 1 $$
$$ S_{n,m-1} = \sum_{k=0}^n k^{m-1} = \frac{1}{C^1_n} \left( (n+1)^m - \sum_{k=0}^n C_n^2 k^{m-2} - ... - 1 \right) \Rightarrow$$
$$ \boxed{ S_{n,m} = \sum_{k=0}^n k^m = \frac{1}{C^1_n} \left( (n+1)^{m+1} - C_n^2 S_{n,m-1} - C_n^3 S_{n,m-2} - ... - C_n^{n-\overline{m-1}} S_{n,m-n +\overline{m-1}+1} - (n + 1) \right) }$$
$$ S_{n,3} = \sum_{k=0}^n k^3 = \frac{1}{4} \left( (n+1)^4 - 6 S_{n,2} - 4 S_{n,1} - (n + 1) \right) = \left( \frac{n(n + 1)}{2} \right)^2 $$
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